Finite Group Databasebeta

← #24919084

Subgroups — #24919084

Order 10624.

556 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 556 conjugacy classes of subgroups — too many to draw. Only its 133 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#14581819C212yesmaximal, complemented
2#19128373C211yesmaximal, characteristic, complemented
3D4×D332C211yesmaximal, characteristic, complemented
4#19128078C211yesmaximal, characteristic, complemented
5#19127955C212yesmaximal, complemented
6#19127901C211yesmaximal, characteristic, complemented
7#19127896C211yesmaximal, characteristic, complemented
8#19127849C212yesmaximal, complemented
9#19127836C212yesmaximal, complemented
10#19127780C211yesmaximal, characteristic, complemented
11C4×SD16×C83C211yesmaximal, characteristic, complemented
12C2×#13050591C2212yescomplemented
13C22×D332—21no—
14C2×D4×D83C2212yes—
15#14581819—28no—
16C2×#13050577C2212yescomplemented
17C2×D664C2212yescomplemented
18C2×D664C2212yescomplemented
19C2×#13050575C2212yes—
20C2×#13050573C2212yescomplemented
21#14581789—21no—
22#14581789C2211yescharacteristic
23#14581789C2211yescharacteristic, complemented
24#14581767—22no—
25#14581767C2212yes—
26#14581767C2212yescomplemented
27#14581766—21no—
28#14581762—22no—
29C4×D332—21no—
30C4×D332C2211yescharacteristic
31C4⋊(C83⋊C8)C2211yescharacteristic, complemented
32#14581712C2212yescomplemented
33#14581712C2212yescomplemented
34(C2×D83)⋊C8C2212yescomplemented
35C2×SD16×C83C2211yescharacteristic
36C4×Q8×C83C2211yescharacteristic, complemented
37C4×D4×C83C2211yescharacteristic
38#106×C83C2211yescharacteristic, complemented
39#103×C83C2211yescharacteristic, complemented
40#102×C83C2211yescharacteristic
41C4×C8×C83C2211yescharacteristic, complemented
42C23×D83—22no—
43#13050591—24no—
44#13050591—42no—
45C2×D332—22no—
46C2×D332—22no—
47C2×D332—21no—
48C2×D332—21no—
49C2×D332—22no—
50C2×D332—21no—
51C2×D332—21no—
52C2×D332—44no—
53C2×D332D411yescharacteristic
54C2×D332C2312yes—
55C2×D332C2312yes—
56C2×C22⋊D83—22no—
57D4×D83—24no—
58D4×D83—42no—
59C2×C4×D83—22no—
60C2×C4×D83C2312yes—
61#13050577—24no—
62D664—24no—
63D664—42no—
64#13050575—24no—
65#13050573—24no—
66C2×C83⋊C8C2312yes—
67C4⋊Dic83D411yescharacteristic
68(C2×D83)⋊C4—22no—
69(C2×D83)⋊C4—22no—
70(C2×D83)⋊C4D412yes—
71C2×Q8×C83C2311yescharacteristic
72C2×D4×C83C2311yescharacteristic
73C22×C4×C83—21no—
74SD16×C83D414yescomplemented
75C2×C8×C83C2311yescharacteristic
76C2×C8×C83C2311yescharacteristic
77C4⋊C4×C83—21no—
78C4⋊C4×C83C2311yescharacteristic
79C4⋊C4×C83C2311yescharacteristic
80C22⋊C4×C83—21no—
81C42×C83—21no—
82C42×C83C2311yescharacteristic
83C23×C83—21no—
84C22×D83—22no—
85C22×D83—22no—
86C22×D83—44no—
87C22×D83—44no—
88C22×D83—82no—
89C22×D83C2×D412yes—
90Q8×C83—21no—
91Q8×C83C2×D412yescomplemented
92D4×C83—21no—
93D4×C83C2×D412yes—
94C2×C4×C83—21no—
95C2×C4×C83—21no—
96C2×C4×C83—21no—
97C2×C4×C83—42no—
98C2×C4×C83C2×D411yescharacteristic
99C2×C4×C83C2×D411yescharacteristic
100C2×C4×C83C2411yescharacteristic
101C22⋊D83—44no—
102C2×Dic83C2×D412yes—
103D332—22no—
104D332—22no—
105D332—22no—
106D332—22no—
107D332—22no—
108D332—44no—
109D332—44no—
110D332—42no—
111D332—42no—
112D332—81no—
113D332—81no—
114D332—82no—
115C4×D83—22no—
116C4×D83—22no—
117C4×D83—44no—
118C8×C83—21no—
119C8×C83C2×D412yescomplemented
120C83⋊C8—22no—
121C22×C83—22no—
122C22×C83—41no—
123C22×C83—41no—
124C22×C83C22×D411yescharacteristic
125C2×D83—22no—
126C2×D83—22no—
127C2×D83—22no—
128C2×D83—42no—
129C2×D83—42no—
130C2×D83—42no—
131C2×D83—42no—
132C2×D83—42no—
133C2×D83—42no—
134C2×D83—88no—
135C4×C83—22no—
136C4×C83—21no—
137C4×C83—41no—
138C4×C83—41no—
139C4×C83AGL(1,ℤ/8)12yescomplemented
140C4×C83C22×D411yescharacteristic
141C4×C83H(5,𝔽2)11yescharacteristic
142Dic83—22no—
143C2×C83—42no—
144C2×C83D4211yescharacteristic
145C2×C83C2×AGL(1,ℤ/8)11yescharacteristic
146C2×C83#57611yescharacteristic
147D83—42no—
148D83—82no—
149D83—82no—
150#3274—831nomaximal, Sylow
151C83#327411yescharacteristic, complemented, Sylow
152C2×AGL(1,ℤ/8)—832no—
153#550—831no—
154D42—831no—
155#493—831no—
156#469—832no—
157#460—831no—
158#459—831no—
159#449—832no—
160#447—832no—
161C4×SD16D8311yescharacteristic, complemented
162#436—831no—
163C2×#41—832no—
164C22×D4—832no—
165C22×D4—1661no—
166AGL(1,ℤ/8)—1668no—
167C2×SD16—832no—
168C2×SD16C2×D8311yescharacteristic
169C2×D8—832no—
170C2×D8—832no—
171C2×D8—832no—
172C2×M16—832no—
173#127—831no—
174#127—831no—
175#127—1661no—
176#121—832no—
177#121—832no—
178#121—1662no—
179#121—1661no—
180#120—1662no—
181C4×Q8C2×D8311yescharacteristic, complemented
182C4×D4—831no—
183C4×D4—1661no—
184C4×D4C2×D8311yescharacteristic
185#106C2×D8311yescharacteristic, complemented
186C4⋊C8—831no—
187#103C2×D8311yescharacteristic, complemented
188#102—832no—
189#102—832no—
190#102C2×D8311yescharacteristic
191C22⋊C8—832no—
192C4×C8C2×D8311yescharacteristic, complemented
193C24—1662no—
194#41—3322no—
195#41—1664no—
196C2×Q8C22×D8311yescharacteristic
197C2×D4—3324no—
198C2×D4—3322no—
199C2×D4—831no—
200C2×D4—832no—
201C2×D4—832no—
202C2×D4—1662no—
203C2×D4—1662no—
204C2×D4—1662no—
205C2×D4—1662no—
206C2×D4—1664no—
207C2×D4—1661no—
208C2×D4—1661no—
209C2×D4—1661no—
210C2×D4—1661no—
211C2×D4C22×D8311yescharacteristic
212C22×C4—21no—
213C22×C4—832no—
214C22×C4—1662no—
215SD16—1664no—
216SD16D33214yescomplemented
217D8—3322no—
218D8—1664no—
219D8—1664no—
220M16—1664no—
221C2×C8—832no—
222C2×C8C22×D8311yescharacteristic
223C2×C8C22×D8311yescharacteristic
224C4⋊C4—21no—
225C4⋊C4—831no—
226C4⋊C4C22×D8311yescharacteristic
227C4⋊C4C22×D8311yescharacteristic
228C22⋊C4—21no—
229C22⋊C4—832no—
230C22⋊C4—1662no—
231C22⋊C4—1662no—
232C42—21no—
233C42C22×D8311yescharacteristic
234C23—21no—
235C23—3324no—
236C23—3324no—
237C23—832no—
238C23—6642no—
239C23—1662no—
240C23—1662no—
241Q8—21no—
242Q8C2×D33212yescomplemented
243D4—21no—
244D4—3324no—
245D4—3324no—
246D4—3324no—
247D4—3322no—
248D4—3322no—
249D4—6641no—
250D4—6641no—
251D4—6642no—
252D4—1662no—
253D4—1662no—
254D4—1662no—
255D4—1662no—
256D4—1662no—
257D4C2×D33212yes—
258C2×C4—21no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—42no—
262C2×C4—3324no—
263C2×C4—832no—
264C2×C4—1662no—
265C2×C4—1662no—
266C2×C4D4×D8311yescharacteristic
267C2×C4D4×D8311yescharacteristic
268C2×C4C23×D8311yescharacteristic
269C8—21no—
270C8—1662no—
271C8C2×D33212yescomplemented
272C22—22no—
273C22—41no—
274C22—41no—
275C22—3322no—
276C22—3322no—
277C22—3322no—
278C22—3322no—
279C22—3322no—
280C22—3322no—
281C22—6648no—
282C22—1662no—
283C22—1662no—
284C22—1662no—
285C22C2×D4×D8311yescharacteristic
286C4—22no—
287C4—21no—
288C4—41no—
289C4—41no—
290C4—1662no—
291C4#1458181912yescomplemented
292C4C22×D33211yescharacteristic
293C4#1458183811yescharacteristic
294C2—42no—
295C2—3322no—
296C2—6642no—
297C2—6642no—
298C2D4×D33211yescharacteristic
299C2C2×#1458181911yescharacteristic
300C2#1912850311yescharacteristic
subgroupG/H
centerC22C2×D4×D83
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
FittingC4×SD16×C83C2
socleC22×C83C22×D4
hypercenterC4×SD16D83
hypocenterC83#3274
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C42 ◁ C4×SD16

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. C2×#14581819C2semidirect
2. #19128373C2semidirect
3. D4×D332C2semidirect
4. #19128078C2semidirect
5. #19127955C2semidirect
6. #19127901C2semidirect
7. #19127896C2semidirect
8. #19127849C2semidirect
9. #19127836C2semidirect
10. #19127780C2semidirect
11. C4×SD16×C83C2semidirect
12. C2×#13050591C22semidirect
16. C2×#13050577C22semidirect
17. C2×D664C22semidirect
18. C2×D664C22semidirect
20. C2×#13050573C22semidirect
23. #14581789C22semidirect
26. #14581767C22semidirect
31. C4⋊(C83⋊C8)C22semidirect
32. #14581712C22semidirect
33. #14581712C22semidirect
34. (C2×D83)⋊C8C22semidirect
36. C4×Q8×C83C22semidirect
38. #106×C83C22semidirect
39. #103×C83C22semidirect
41. C4×C8×C83C22semidirect
74. SD16×C83D4semidirect
91. Q8×C83C2×D4semidirect
119. C8×C83C2×D4semidirect
139. C4×C83AGL(1,ℤ/8)semidirect
151. C83#3274semidirect
161. C4×SD16D83semidirect
181. C4×Q8C2×D83semidirect
185. #106C2×D83semidirect
187. #103C2×D83semidirect
192. C4×C8C2×D83semidirect
216. SD16D332semidirect
242. Q8C2×D332semidirect
271. C8C2×D332semidirect
291. C4#14581819semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×#145818192
#191283732
D4×D3322
#191280782
#191279552
#191279012
#191278962
#191278492
#191278362
#191277802
C4×SD16×C832
#327483